1. Introduction

The internal density profile and shape of Earth is commonly studied by indirect physical methods, starting from the 1793 deduction by Cavendish that the Earth must have a dense core, based on a gravitational calculation, to the current methods which use sismic wave propagation, studies of the vibrational modes of the Earth as an elastic body, or temperature constraints. However, these studies are still subject to intrinsic ambiguities (Aki and Richards, 1980; Lay and Wallace, 1995). For these reasons, independent measurements of the density profile would be of considerable value and in the last years complementary methods for Earth radiography were pursued, by using cosmic beams of neutrinos and muons. The principle of tomography by energetic beams is essentially the same as X-ray tomography, except for substituting penetrating particles in place of X-rays. By measuring their absorption along different paths through a solid body, one can deduce the nucleon density in the interior of the object.

The Earth’s tomography with ultra-high energy cosmic neutrinos seems to provide a viable independent determination of the Earth’s internal structure (Jain et al., 1999; Reynoso and Sampayo, 2004), giving some information on its internal density distribution. On the other side, cosmic ray muon radiography can be applied to km-size objects located at elevations above where the detector is placed (Ambrosio et al., 1995; Nagamine et al., 1995; Tanaka et al., 2003, 2005, 2007a–c, 2008; Tanaka and Yokoyama, 2008).

The idea of neutrino tomography is based on the fact that Earth becomes opaque to neutrinos of energy exceeding ∼10 TeV, since at this energy the diameter of Earth corresponds to about one absorption length. Such neutrinos are produced by collisions of cosmic rays with the Earth atmosphere, and interact during their travel in such a way that the charged leptons produced, essentially muons, could emerge from the surface and be detected by a km3 Neutrino Telescope (NT). In this concern, after the first generation of telescopes which has proved the feasibility of the Cerenkov detection technics under deep water (Balkanov et al., 1999) and ice (Ahrens et al., 2002) by detecting atmospheric neutrinos, we are likely approaching the first detections of astrophysical neutrinos at the IceCube (Ahrens et al., 2004) telescope, being completed at the South Pole, and possibly at the smaller ANTARES (Spurio, 2006) telescope under construction in the Mediterranean. Moreover, ANTARES as well as NESTOR (Aggouras et al., 2006) and NEMO (Migneco et al., 2008) are involved in R&D projects aimed at the construction of a km3 NT in the deep water of the Mediterranean sea, coordinated in the European network KM3NeT (Katz, 2006).

On the other side, cosmic ray muons are also generated from cosmic rays in the atmosphere and arrive at angles ranging from vertical to horizontal (Thompson and Whalley, 1975) with a smaller number of neutrino-induced muons directed upward. These particles are highly penetrating and a typical horizontally-arriving cosmic-ray muon with an energy of 1 TeV penetrates 2.6 km of water. Thus, cosmic ray muon radiography can be applied to Earth structures with size of the order of a km, placing a detector with a smaller area than a NT (∼m2) near a volume that is higher in elevation adjacent to the detector.

In both of these technics, full Monte Carlo simulations are needed for the extraction of physical parameters from the detected numbers of events. In the following we will report on the status of studies, aiming to simulate the interaction of neutrinos inside the Earth and the propagation of their parents leptons inside matter.

2. Earth’s Tomography with Neutrinos and Muons

In order to understand how the number of charged lepton events at a km3 NT depends on the density of matter crossed by HE neutrinos, let us remind the formalism developed in Cuoco et al. (2007). In the following we will refer our calculation to a km3 NT placed at NEMO site.

We define the km3 NT fiducial volume as that bounded by the six lateral surfaces Σ a (the subindex a = D, U, S, N, W, and E labels each surface through its orientation: Down, Up, South, North,West, and East), and indicate with Ω a ≡ (θ a , φ a ) the generic direction of a track entering the surface Σ a . The scheme of the NT fiducial volume and two examples of incoming tracks are shown in Fig. 1.

Fig. 1
figure 1

The angle definition and the fiducial volume of a km3 NT.

Full size image

Let dΦ v /(dE v dΩ a ) be the differential flux of UHE v μ + v μ . The number per unit time of μ leptons emerging from the Earth surface and entering the NT with energy E τ is given by

(1)

The kernel is the probability that an incoming v μ crossing the Earth, with energy E v and direction Ω a , produces a μ-lepton which enters the NT fiducial volume through the lateral surface dSa at the position with energy E μ (see Fig. 1 for the angle definition).

As already shown in Cuoco et al. (2007) (the interested reader can find in Miele et al. (2006) more details on the equations reported in the following), a typical event corresponds to the simultaneous fulfillment of the following conditions:

  1. 1)

    A v μ with energy E z travels over a distance z through the Earth before interacting. The corresponding probability P1 is given by

    (2)

    with

    (3)

    where NA is the Avogadro number and ϱr is the Earth density assumed to be constant.

  2. 2)

    The neutrino produces a μ in the interval z, z + dz, the probability of such an event being

    (4)
  3. 3)

    The produced μ emerges from the Earth rock with anenergy E′ μ . This happens with a probability

    (5)
  4. 4)

    Finally, the μ lepton emerging from the Earth rock propagates in water and enters the NT fiducial volume through the lateral surface Σ a at the point r⃗ a with energy E μ . The corresponding survival probability is

    (6)

    where ϱw stands for the water density and z w (r⃗ a ,Ω a ) represents the total length in water before arriving to the fiducial volume for a given track entering the lateral surface Σ a at the point r⃗ a and with direction μ a .

Collecting together the different probabilities in Eqs. (2), (4), (5) and (6), we have

(7)

Now, the Earth density profile enters via the matter density, ϱr, in most of the previous expressions, and thus one must expect a certain sensitivity of the number of detected events to the characteristic of the Earth model.

Similar considerations apply to the case of muon radiography, with the difference that now the primary particles considered are the muons produced through the cosmic ray interaction with the Earth atmosphere. Such muons enter Earth and lose energy by ionization and radiative processes: bremsstrahlung, direct production of e+e- pairs, and photonuclear interaction. The total muon energy loss may be expressed as a function of the amount of matter traversed as

(8)

where α is the ionization loss and β is the fractional energy loss by the three radiation processes (Gaisser and Stanev, 2008). These parameters are quite sensitive to the chemical composition of the rock.

3. Simulation of Neutrino and Lepton Propagation Inside Matter

Our work aims to the development of Monte Carlo methods for simulating neutrino and muon interaction in matter. A possible line towards this purpose is based on the combination of two existing Monte Carlo, the HERWIG hadron generator (Corcella et al., 2001) to simulate neutrino interactions inside the Earth, and GEANT4 (Geometry ANd Tracking) (Agostinelli et al., 2003; Allison et al., 2006, http://www.geant4.org) to simulate the propagation of the produced muons in matter.

HERWIG (Corcella et al., 2001) is an event generator for high-energy processes particularly suited for detailed simulation of QCD parton showers. It provides simulation of hard lepton-lepton, lepton-hadron and hadron-hadron scattering and soft hadron-hadron collisions within a single package. In particular, HERWIG can simulate the neutrino interaction and, in a previous work (Ambrosio et al., 2003), it was already combined with the Monte Carlo CORSIKA (Heck et al., 1998), by some of the present authors, to develop a new version of CORSIKA capable of simulating atmospheric showers induced by neutrinos.

Geant4 is a toolkit for the simulation of the passage of particles through matter. Its areas of application include high energy, nuclear and accelerator physics, as well as studies in medical and space science.

In the simulations here described, we have generated a large number of tracks crossing the NEMO site by means of a detailed Digital Elevation Map of the under-water Earth surface, which is available from the Global Relief Data survey (ETOPO2): a grid of altimetry measurements with a vertical resolution of 1 m averaged over cells of 2 minutes of latitude and longitude. The Earth is described by a simplified version of the Preliminary Reference Earth Model (PREM) (Dziewonski, 1971), with three density regions: crust, mantle, and core.

We first describe the characteristic of the HERWIG simulation. In Fig. 2 we show from left to right the outgoing muon energy fraction and the multiplicity distribution of the first interaction particles, respectively, for the Charge Current (CC) interaction of 2000 neutrino events simulated by HERWIG at the energy of 104 GeV, while last plot on the right show the type of particles produced.

Fig. 2
figure 2

From left to right, the plots show the outgoing muon energy fraction and the multiplicity distribution of the first interaction particles, respectively, for the Charge Current (CC) interaction of 2000 neutrino events simulated by HERWIG at the energy of 104 GeV, and finally the type of particles produced at the energy of 105 GeV.

Full size image

The first interaction particles produced by HERWIG are extracted and injected in subsequent runs of GEANT4. The GEANT4 simulation is performed following particle tracking in a material with the rock characteristic, divided into a maximum of 5 different zones with 3 possible densities (see Fig. 3). This corresponds to three different kind of neutrino tracks inside the Earth: the ones which go 1) through the core, 2) through mantle and crust, and 3) only through the crust. The tracking stops if the track leaves the tracker (track A in Fig. 3) or if the energy of the particle is less than 1 GeV (track B in Fig. 3) so to make the simulation faster and take into account only interesting particles.

Fig. 3
figure 3

Type of tracks considered in the simulation with GEANT4.

Full size image

In order to test the results coming from the link of HERWIG and GEANT4, we developed also a simplified Monte Carlo, which simulates neutrino interaction in Earth, and propagates the outgoing muon, taking into account the phenomenon of neutrino regeneration by NC interaction and muon energy loss in matter. The number of neutrinos injected at each angular bin was scaled according to the known flux of atmospheric neutrinos, depending on energy and angle. An energy threshold was applied to the muons detected in the fiducial volume.

To simulate the radiography of Earth structures by cosmic ray muons, a beam of muons is tracked while crossing the rock of the considered geological structure. Since the dimensions of the detector are orders of magnitude smaller than the dimensions of the structure that we want to investigate, in first approximation we can consider the detector as pointlike and, by using a Digital Elevation Map, for each direction crossing the detector we can evaluate the amount of rock crossed by the muon. In such a way, as in the case of neutrino Earth radiography, we can define the dimensions of the passive material where GEANT4 performs the tracking. To study the presence of holes or of zones with a different density in the passive material structure, one or more substructure can be added. The output gives information about the outgoing muon energy and direction.

4. Results and Conclusions

In Fig. 4 we show number and type of the particles arriving to the NT in the first 2000 events of a GEANT4 run for a primary neutrino of energy 104 GeV: the plot on the left is reported as a contour on the right, where the more large are the circles, the higher is the corresponding number of particles. Moreover, the particles corresponding to some of the particle codes are highlighted. On the other side, Fig. 5 shows the angular distribution of events simulated with our simplified Monte Carlo, and is in nice agreement with the line corresponding to Eth = 1 TeV in figure 1(a) of González-García et al. (2008).

Fig. 4
figure 4

Number and type of the particles arriving to a NT in the first 2000 events of a GEANT4 run for a primary neutrino of energy 104 GeV (see text).

Full size image
Fig. 5
figure 5

Zenith angle distribution of neutrino induced events in a NT for a muon energy threshold of 1 TeV and the PREM. Neutrinos coming from below correspond to θ = 0.

Full size image

Concerning the muon radiography of geological structures, as first step of our study we performed a very preliminary analysis: we simulated 10000 muons with an energy of 1 TeV, in the z direction, and tracked them trough 1 km of rock with density 2.6 g/cm3. Only 3746 of these muons were able to cross the whole structure and their angular resolution was ∼10 mrad, as shown in Fig. 6. In Fig. 7 the momentum of the outgoing muon in the x and y direction and the resolution obtained as a function of the outgoing momentum are shown, respectively. As we can see, the largest contribution to the resolution is due to muons with a momentum below 40 GeV.

Fig. 6
figure 6

Angular resolution of the outgoing muons surviving 1 km of rock with density 2.6 g/cm3.

Full size image
Fig. 7
figure 7

Outgoing muon momentum in the x and y directions (left) and the resolution as a function of the outgoing momentum (right).

Full size image

By summarizing, the results we have presented confirm that neutrino or muon radiography of geological structures is a very promising research field, and with the help of the simulations we are developing one can achieve the final goal of recognizing a non trivial density profile with a good level of statistical confidence.